= Solution
Writing $x_i=(x_i,y_i,z_i)$, rotation about the $z$-axis gives particle $i$ the <speed> $\omega\sqrt{x_i^2+y_i^2}$. Its <kinetic energy> is therefore $m_i\omega^2(x_i^2+y_i^2)/2$, so
$$
T=\frac12I\omega^2,
\qquad
I=\sum_{i=1}^Nm_i(x_i^2+y_i^2).
$$
This is the defining mass sum for the <moment of inertia> about the $z$-axis.
For the uniform cuboid, the <mass density> is $\rho=M/(8abc)$ and
$$
I=\rho\int_{-a}^a\int_{-b}^b\int_{-c}^c(x^2+y^2)\,dz\,dy\,dx
=\boxed{\frac M3(a^2+b^2)}.
$$
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